We consider a network of N symmetrically interconnected neurons subject to additive as well as multiplicative (in the synaptic connections) noise. In a previous paper, the dynamics of a single neuron, in the presence of additive and multiplicative noise, was obtained through the procedure of adiabatic elimination. We analyze the macroscopic potential that describes the steady-state properties of this effective neuron, considering the effects of additive as well as multiplicative noise (in the synaptic connections).

The quantum Brownian particle, immersed in a heat bath, is described by a statistical operator whose evolution is ruled by a generalized master equation (GME). The heat bath’s degrees of freedom are considered to be either white-noise or colored-noise correlated, while the GME is considered under either the Markov or non-Markov approaches. The comparisons between these considerations are fully developed, and their physical meaning is discussed.

A stochastic Markov process is defined which generalizes a class of processes previously considered by the author [Phys. Rev. A 42, 4485 (1990)]. The enlarged class unifies such apparently unrelated processes as the Ornstein-Uhlenbeck process, the generalized Verhulst-Landau processes, the generalized Rayleigh process, the hyperbolic tangent processes, and many other cases of physico-mathematical interest. The Fokker-Planck equation associated with this unifying stochastic process is solved analytically for the transition probability density function, using a similar constructive solution method as in the author’s previous work. The result is obtained as an eigenfunction expansion over a generally mixed spectrum. The discrete eigenfunctions are related (but not identical) to Jacobi polynomials. It is shown that the results for the above-mentioned processes are obtainable as limiting cases, and that some additional solvable equivalent Schrödinger problems can be defined.

We investigate the effects of long-range correlated noise on the dynamics of a growing interface in which the total volume under the interface is conserved. For a nonconserved noise the model corresponds to the nonlinear molecular-beam-epitaxy model with spatially correlated noise. We use both a dynamic renormalization-group approach and a Kolmogorov type of scaling applied to the Langevin equation describing the model, to study the dynamical evolution of the interface in arbitrary interface dimension d’. We find that for d’<2 any amount of spatial correlation is relevant and leads to a universality class different from that of the uncorrelated white noise. Both methods are found to give the same expressions for the growth exponents α and β as a function of the noise parameter and the spatial dimension of the system. These exponents are expected to be exact because of nonrenormalization of the spectral density function and the vertex function.

The variational approximation to the thermal properties of a quantum-mechanical system proposed by Giachetti and Tognetti [Phys. Rev. Lett. 55, 912 (1985); Phys. Rev. B 33, 7647 (1986)], Feynman and Kleinert [Phys. Rev. A 34, 5080 (1986)], and others is taken as the starting point, and a systematic perturbation series is developed for both the free energy and the particle density. Generalizations of the usual Feynman propagator appear naturally in the formalism. The anharmonic oscillator is treated as an example; the second-order correction to the free energy and the first-order correction to the particle density are evaluated explicitly and shown to lead to significant improvements over the variational solution.

We report some unusual dynamical phenomena found in a Hamiltonian system of 1.5 degrees of freedom. The Hamiltonian describes the guiding-center motion of a charged particle in the presence of three electrostatic waves. There is a collapse of two hyperbolic fixed points into one, which gives rise to a mechanism for the onset of large-scale stochasticity. A nonlinear stability analysis demonstrates the existence of three invariant manifolds at the collapsed point. With increasing perturbation the large-scale chaotic motion exhibits anomalous diffusion with a ${\mathit{t}}^{1/3}$ growth of the variance, which is explained by trapping in a hierarchy of nested cantori.

We use the model of the adsorption of gas molecules to a surface treated as a two-dimensional Ising lattice gas to illustrate the use of equilibrium statistical mechanics (specifically, equilibrium activity series) to construct the kinetics of the adsorption in the limit that surface diffusion is rapid with respect to the rate of adsorption (assumption of internal equilibrium). These conditions generally give an upper bound to the rate of relaxation relative to a system with a finite or zero rate of diffusion on the surface. For the plane-square lattice and special choices of the kinetic parameters one can use known 15-term equilibrium activity series to give 15 terms of the power series in the time, or the inverse of this function. At the critical point the relaxation is extremely slow, varying as ${\mathit{t}}^{\mathrm{-}1/14}$. We find that if one expresses the kinetics in terms of a time-dependent activity the resulting series are much better behaved than the corresponding density series.

Velocities and trajectories of interacting defects in the real Ginzburg-Landau equation are obtained by applying equations of motion of defects under the influence of the local phase field in conjunction with the Green’s-function solution of the heat equation governing the phase field generated by moving defects.

The working hypotheses of the fixed-scale transformation as applied to the percolation problem in two dimensions are checked with Monte Carlo simulations. The set of points given by the intersection of a line and the infinite cluster is correctly described by a random Cantor set built with the generators proposed in the theory. But the matrix of the fixed-scale transformation that describes growth at the same scale is not scale independent. A simple explanation is given for this behavior. Keeping the spirit of the theory, it is proposed to coarse grain the infinite cluster with a two-dimensional random Cantor set. In this case, the matrix describing the transitions with respect to growth between the different diagrams is scale independent and is not a unit matrix. One can in principle extract the fractal dimension from it.

A real-space renormalization scheme for reaction-limited cluster-cluster aggregation is presented based on the properties of the reaction surface. For the soluble monodisperse case both the fractal dimensions and reaction surface exponents can be calculated in good agreement with simulation and experiment. The associated scaling functions are also found. Taking the limits d=1 and d→∞ in the scheme, we recover linear and ghost aggregates exactly as expected. The polydisperse case is also studied in some detail, although a numerical solution has not been obtained.

A model is defined that interpolates between integer dimension and integer q-state Potts models. Phase transitions in these models are studied, and in particular the critical value ${\mathit{q}}_{\mathit{c}}$, where the phase transition changes from second to first order, is determined for several dimensions. For (d,${\mathit{q}}_{\mathit{c}}$) the values (2.0,4.05), (2.5,2.68), (3.0,2.21), (3.5,2.15), and (4.0,2.07) are obtained.

The free energies for the oscillators with quartic, sextic, and mixed quartic-sextic anharmonicities have been calculated over a wide range of anharmonicity parameter and of temperature employing the approximation methods put forward by Büttner and Flytzanis [Phys. Rev. A 36, 3443 (1987)] and by Feynman and Kleinert [Phys. Rev. A 34, 5080 (1986)]. A comparison of these values with the exact ones shows that the prescription given by Feynman and Kleinert is far superior, and that it produces quite accurate results for the potentials and temperatures considered here. The success of the Büttner-Flytzanis technique at 0 K and its failure at the intermediate as well as the high temperatures have also been analyzed.

The renormalized hypervirial-Padé scheme has been employed to determine energy eigenvalues of a double-well oscillator with mixed quartic and sextic anharmonicities and the special cases obtained from its potential by taking either anharmonicity as zero. The renormalization parameters calculated from two analytical expressions (one of these being suggested as an improvement of the earlier prescription) are found to be inadequate, while the approach in which this parameter is treated as a variable quantity works well. The results for a wide range of anharmonicity values are compared with the findings of other workers and the agreement is good to superb. It turns out that the technique applied here is effectual when the wells are not very deep. Discussion of the dependence of the renormalization parameter and the energies on the quantities defining the double-well potential and also on the quantum number is also included.

We investigate some aspects of q Heisenberg algebra. We show how su(2) and su(1,1) generators can be constructed in terms of the q creation and annihilation operators. We also construct the coherent states for the q oscillator and show that they can be obtained by the action of a displacement operator on the vacuum. For the multimode case with q=0, corresponding to the infinite statistics of Greenberg [Phys. Rev. Lett. 64, 705 (1990)], we generalize our single-mode construction to obtain the corresponding coherent states. These states, which are eigenstates of the annihilation operators, interestingly, turn out to be degenerate owing to the noncommutativity of the displacement operators.

We show that a single para-Bose oscillator may be regarded as a deformed Bose oscillator. We construct a nonlinear realization of the single-mode para-Bose algebra in terms of a single boson. This is in contrast to the Green decomposition that expresses a single para-Bose oscillator in terms of p anticommuting bosons. We also construct an operator canonically conjugate to the para-Bose annihilation operator that permits us to carry over familiar constructions to the para-Bosonic case.

Colored-noise-driven bistable systems are considered in the weak-noise limit D→0 and long-correlation-time limit τ→∞. The analytic solution of the two-dimensional Fokker-Planck equation is worked out. The theoretical result is compared with a numerical simulation.

Stochastic resonance is a cooperative effect of noise and periodic driving in bistable systems. It can be used for the detection and amplification of weak signals embedded within a large noise background. In doing so, the noise triggers the transfer of power to the signal. In this paper we first present general properties of periodically driven Brownian motion, such as the long-time behavior of correlation functions and the existence of a ‘‘supersymmetric’’ partner system. Within the framework of nonstationary stochastic processes, we present a careful numerical study of the stochastic resonance effect, without restrictions on the modulation amplitude and frequency. In particular, in the regime of intermediate driving frequencies which has not yet been covered by theories, we have discovered a secondary resonance at smaller values of the noise strength.

The scaling behavior of the additive random-matrix model H=(${\mathit{H}}^0$+λV)/(1+${\lambda }^2$${)}^{1/2}$ has been investigated. This model, capable of producing transitions between different degrees of level repulsion for the eigenvalues, has been analyzed for the case of the transition from the Poisson distribution exp(-S) of the distances between adjacent eigenvalues to the Wigner distribution (πS/2)exp(-${\mathit{S}}^2$π/4). We propose expressions for the level-spacing distribution and the distribution of the eigenvector components and demonstrate their reliability by means of numerical computations. For the eigenvalues, the transition proceeds at a rate that scales as the square root of the matrix dimension. For the eigenvalues, the rate is independent of this dimension.

The kinetic theory for lattice-gas cellular automata is extended beyond Boltzmann’s mean-field approximation by including correlated ring-type collisions. This theory provides explicit expressions for time-correlation functions for all times in terms of a ring collision integral. We also obtain some exact results for time-correlation functions for short times. In particular, deviations from the Boltzmann equation result observed in computer simulations of the velocity-autocorrelation function of three-dimensional systems after two time steps are explained. For long times the results obtained reduce to those found using phenomenological mode-coupling theory.

We present Rayleigh-Brillouin scattering results of ${\mathrm{H}}_2$+Ar, ${\mathrm{H}}_2$+Xe, and He+Xe mixtures that support the existence of fast and slow sound in binary gas mixtures. An experimental dispersion relation of both the fast and slow sound modes is given. The approach to the Gaussian limit is also investigated. An alternative characteristic length scale is used as the scaling parameter of the sound modes. This length scale proved to be a proper choice as a scaling parameter of the dispersion relation of different gas mixtures.